"Straight" is poorly defined and so the original question (the intention, if I understood it) has no correct answer. It would depend on how you define a straight path and in what frame of reference.
Due to the effect of cetrifugal force, do you weigh more at the poles than you do at the equator?
If you mean the acceleration you feel toward the center of the earth then yes. However, what you will perceive as "down" direction will not point toward the center, except exactly on the equator and the poles. The ratio between the centrifugal acceleration and gravity is of the order 10^-20, so you are not likely to feel this (punched with my thick finger on the calculator so I might be off).
Here is another funny tidbit. When you are running, you are "lighter" when you run east, then when you run west.
Okay, so what about this....if two planes were traveling, one directly over the other....the first at 500 feet and the second at 50000 feet. If both of the planes had to travel from point A to point B staying at the exact orientation throughout. Both starting over point A and finishing over B at the same time....would the higher one have to travel faster to reach point B at the same time? Because of the curvature of the earth, wouldn't he have to cover more distance in the same amount of time?? The same thing with a merry-go-round. The outside seats would go faster because of the same principle....right?
Yes, the arc is longer due to longer radius. However, consider this: the length of the arc is proportional to the radius, being the radius of the earth (~6400 km) plus the altidude (500 ft ~ 0, 50kft ~ 17 km), so the ratio between the distances the planes travel is ~(6400+17)/6400 = 1.003. that is less than 1/3 of a percent. If the lower one flies 10,000 km, the higher one will have to fly about 30 km more. For a big jet this is about as big as its holding pattern. Also, he has to climb 17 km and then decent 17 km. That is already more than the difference due to the curvature.
